Finite Group Databasebeta

← #27274646

Subgroups — #27274646

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810695C211yesmaximal, characteristic
2#20274493C211yesmaximal, characteristic, complemented
3C22⋊C4×#88322C211yesmaximal, characteristic, complemented
4#20273033C211yesmaximal, characteristic
5#20272563C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C4×#99×C53C211yesmaximal, characteristic
8C22×#11719917C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16810695—21no—
11#16810695—21no—
12#16810695—21no—
13#16810695—21no—
14C2×C4⋊#88322C411yescharacteristic, complemented
15C2×C22⋊#88322C411yescharacteristic, complemented
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×#13133128—21no—
20C2×#13133128C411yescharacteristic, complemented
21C2×C4×#88322—21no—
22#16810378—21no—
23#16810378—21no—
24#16810349C412yescomplemented
25#16810347C412yescomplemented
26C2×C4×D4×C53C2211yescharacteristic
27C2×#99×C53C412yes—
28C4×C22⋊C4×C53C2211yescharacteristic
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#342C412yes—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—42no—
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2×C411yescharacteristic
39C22×#88322C2×C411yescharacteristic
40C22×#88322C2×C411yescharacteristic
41C22×#88322D411yescharacteristic
42C22×#88322D411yescharacteristic
43C22×#88322D411yescharacteristic
44C22×#88322D411yescharacteristic
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322C2×C412yes—
54C22⋊#88322C2×C412yes—
55#13133128—21no—
56#13133128—21no—
57#13133128—42no—
58C4×#88322—42no—
59C22×D4×C53C2×C411yescharacteristic
60C23×C4×C53C2×C411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53C2×C412yes—
68C2×C22⋊C4×C53C2×C412yes—
69C2×C22⋊C4×C53C2×C411yescharacteristic
70C2×C22⋊C4×C53C2×C411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53C2×C411yescharacteristic
74#99×C53—24no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53C4211yescharacteristic
78C24×C53C22×C411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C4211yescharacteristic
92C22×C4×C53C4211yescharacteristic
93C22×C4×C53C4211yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C22⋊C4×C53—21no—
103C22⋊C4×C53—21no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53M1614yescomplemented
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—41no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53—41no—
116#11719917—42no—
117#11719917—42no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322C22⋊C411yescharacteristic
135C2×#88322C22⋊C411yescharacteristic
136C2×#88322C22⋊C411yescharacteristic
137C2×#88322C22⋊C411yescharacteristic
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C2×C4211yescharacteristic
146C23×C53C4×D411yescharacteristic
147C23×C53C4×D411yescharacteristic
148C23×C53C2×M1612yes—
149C23×C53#13112yes—
150D4×C53—42no—
151D4×C53—42no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—24no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53#11711yescharacteristic
179C2×C4×C53C4×D411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181#88322—21no—
182#88322—21no—
183#88322—21no—
184#88322—42no—
185#88322—42no—
186#88322—41no—
187#88322—41no—
188#88322#9912yescomplemented
189#8161—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—42no—
197C22×C53—42no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53C4×C22⋊C411yescharacteristic
204C22×C53C4×M1611yescharacteristic
205C22×C53#40511yescharacteristic
206C22×C53#43512yes—
207C22×C53#43612yes—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—44no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53—82no—
216C4×C53C2×#9911yescharacteristic
217C4×C53#41011yescharacteristic
218C2×#435—531no—
219C4×#99D5311yescharacteristic
220#1752—531no—
221C8×C22⋊C4—531no—
222#1459—531no—
223#1282—531no—
224#1282—531no—
225C2×C53—22no—
226C2×C53—22no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—81no—
230C2×C53#175311yescharacteristic
231C2×C53C4×#9911yescharacteristic
232C2×C53#175711yescharacteristic
233C22×M16—531no—
234C23×C8—531no—
235C2×C4×D4C2×D5311yescharacteristic
236#435—1061no—
237#435—1061no—
238#435—1061no—
239#435—1061no—
240C2×C4⋊C8—531no—
241C2×#99Dic5312yes—
242C2×C22⋊C8—531no—
243C2×C22⋊C8—531no—
244C2×C22⋊C8—531no—
245C2×C22⋊C8—531no—
246C2×#97—531no—
247C2×#97—1061no—
248C2×C4×C8—1061no—
249C4×C22⋊C4C2×D5311yescharacteristic
250C4×C22⋊C4C2×D5311yescharacteristic
251#342Dic5312yes—
252#336—1061no—
253#336—1061no—
254#324—532no—
255#323—532no—
256C53#816111yescharacteristic, complemented, Sylow
257C22×D4C2×Dic5311yescharacteristic
258C23×C4C4×D5311yescharacteristic
259C23×C4C22×D5311yescharacteristic
260C2×M16—1061no—
261C2×M16—1061no—
262C2×M16—2122no—
263C22×C8—531no—
264C22×C8—531no—
265C22×C8—531no—
266C22×C8—531no—
267C22×C8—531no—
268C22×C8—531no—
269C22×C8—531no—
270C22×C8—531no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8—2122no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C2×C4⋊C4C4×D5311yescharacteristic
279C2×C22⋊C4C4×D5311yescharacteristic
280C2×C22⋊C4C2×Dic5312yes—
281C2×C22⋊C4C2×Dic5311yescharacteristic
282C2×C22⋊C4C2×Dic5312yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C4×D5311yescharacteristic
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288#99—24no—
289C22⋊C8—532no—
290C22⋊C8—532no—
291C22⋊C8—1061no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297#97—1061no—
298#97—1061no—
299#97—2122no—
300C4×C8—2122no—
301#95—21no—
302#95—21no—
303C24C4×Dic5311yescharacteristic
304C24C22×Dic5311yescharacteristic
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C4×Dic5311yescharacteristic
318C22×C4C4×Dic5311yescharacteristic
319C22×C4C4×Dic5311yescharacteristic
320C22×C4C2×C4×D5311yescharacteristic
321C22×C4C2×C4×D5311yescharacteristic
322C22×C4D4×D5311yescharacteristic
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4#1171994711yescharacteristic
326M16—2122no—
327M16—2122no—
328C2×C8—531no—
329C2×C8—531no—
330C2×C8—531no—
331C2×C8—531no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8—1062no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—22no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4#1171991714yescomplemented
358C42—21no—
359C42—21no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C2×C4×Dic5311yescharacteristic
372C23D4×Dic5311yescharacteristic
373C23D4×Dic5311yescharacteristic
374C23C2×#1171991712yes—
375C23#1313322412yes—
376D4—42no—
377D4—42no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4C22⋊C4×D5311yescharacteristic
404C2×C4#1313317311yescharacteristic
405C2×C4#1313317711yescharacteristic
406C2×C4#1313317711yescharacteristic
407C8—532no—
408C8—1061no—
409C8—1061no—
410C8—1061no—
411C8—2122no—
412C8—2122no—
413C8—2121no—
414C8—2121no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22C22⋊C4×Dic5311yescharacteristic
429C22C4×#1171991711yescharacteristic
430C22#1681057911yescharacteristic
431C22#1681069512yes—
432C22#1681070112yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4#99×D5311yescharacteristic
442C4#1681059511yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—81no—
448C2#2027449911yescharacteristic
449C2#99×Dic5311yescharacteristic
450C2#2027451711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C4×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC4×#99D53
hypocenterC53#8161
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#99

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20274493C2semidirect
3. C22⋊C4×#88322C2semidirect
5. #20272563C2semidirect
6. #20272563C2semidirect
8. C22×#11719917C4semidirect
14. C2×C4⋊#88322C4semidirect
15. C2×C22⋊#88322C4semidirect
20. C2×#13133128C4semidirect
24. #16810349C4semidirect
25. #16810347C4semidirect
109. C22⋊C4×C53M16semidirect
188. #88322#99semidirect
256. C53#8161semidirect
357. C22⋊C4#11719917semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168106952
#202744932
C22⋊C4×#883222
#202730332
#202725632
C4×#99×C532
#816153